4 Integral calculus

We now turn from differentiation to the other crucial ingredient of the infinitesimal calculus, namely integration. This may be approached in two ways: either as the inverse process of differentiation; or as the means of calculating the area under a given curve, using an argument that serves as a template for many other important applications. However, before discussing these, we must develop the above two approaches, and the relations between them.

4.1 Indefinite integrals

Given a function f(x), the indefinite integral F(x) is defined as the most general solution of the equation

(4.1) Unnumbered Display Equation

It is not unique. Suppose we have a particular solution F0(x), with

(4.2) Unnumbered Display Equation

Then the most general solution can be written

(4.3) Unnumbered Display Equation

where G(x) is any function such that (4.1) is satisfied. On substituting (4.3) into (4.1) and using (4.2), one obtains , and so G(x) = c, where c is a constant. Hence the indefinite integral is given by

(4.4) Unnumbered Display Equation

where F0(x) is any particular solution ...

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